We are alternating Tuesday afternoons with the computer algebra group. We shall have room for two talks every second week. This week there will only be one talk.

Abstract: This result roughly states that for "well represented" maps on a surface, the chromatic number of the underlying graph mainly depends on the map's "local structure".

More presicely, a (voltage) {\it tension} on a directed graph $G$ is a function $f : E(G) -> R$ where, for any circuit C,the accumulated voltage

f(C) = \sum {f(e): e is directed clockwise around C}

-\sum {f(e): e is directed anticlockwise around C}

equals zero. When G is a map on a surface S, we may relax this condition, requiring that f(C) equal zero for every "contractable" circuit C. Such a function f is called a {\it local tension}. Suppose that every noncontractable circuit in G has length at least K. Then, if there exists a local tension

f: E(G) -> [ 1, r ],

then there exists a (ordinary) tension

f: E(G) -> [ 1, r+\epsilon ]

where \epsilon depends only on K and the surface S. This, in turn, implies bounds that the circular chromatic number of G is at most $ r + 1 + \epsilon$. This is joint work with M. DeVos, B. Mohar, D. Vertigan and X. Zhu done during the PiMS Summer Workshop.